Individual, mean, or total?

◈ 6 cards

Three questions, three distributions: one value N(μ, σ²), a mean of n N(μ, σ²/n), a total of n N(nμ, nσ²) — the SD is σ, σ/√n or √n·σ.

The same population, three random variables

Northfield Credit's invoices are $X \sim N (120, 1 5^{2})$ . Three questions about the same day's work:

What is the probability that one invoice exceeds \$132?
What is the probability that the average of 25 invoices exceeds \$132?
What is the probability that the total of 25 invoices exceeds \$3,150?

Each is about a different random variable, with a different SD. The mean is the only thing shared between the first two — and the third has a different mean as well.

One value: $X \sim N (μ, σ^{2})$

Module 7, unchanged. $z = (132 - 120) /15 = 0.80$ , so

$P (X > 132) = 1 - Φ (0.80) = 1 - 0.7881 = 0.2119$

One invoice in five is above \$132.

A mean of $n$ : $\overset{ˉ}{X} \sim N (μ, σ^{2} / n)$

L8.3: the population is normal, so $\overset{ˉ}{X}$ is normal with $SE = 15/ 25 = 3$ .

$z = \frac{132 - 120}{3} = 4.00, P (\overset{ˉ}{X} > 132) = 1 - Φ (4.00) \approx 0$

The table stops at 3.4 and $Φ (4.00) = 0.99997$ ; to four decimals the probability is 0.0000. A single invoice above \$132 is routine; an average of 25 above \$132 essentially never happens, because averaging 25 invoices divides the spread by 5. The two answers differ by a factor of thousands from one change of word.

A total of $n$ : $T \sim N (n μ, n σ^{2})$

The total $T = X_{1} + \dots + X_{n}$ is Module 6.3's sum: means add to $n μ$ , variances add to $n σ^{2}$ , and the SD is $n σ$ — not $nσ$ . For 25 invoices,

$T \sim N (25 \times 120, 25 \times 225) = N (3, 000, 7 5^{2}), SD (T) = 25 \times 15 = 75$

$z = \frac{3 , 150 - 3 , 000}{75} = 2.00, P (T > 3, 150) = 1 - Φ (2.00) = 0.0228$

The total question is the mean question in disguise: $T > 3, 150$ is the same event as $\overset{ˉ}{X} > 126$ , and $z = (126 - 120) /3 = 2.00$ either way. Use whichever form the numbers are given in, but check the two agree when unsure.

The rule

The question is about	Distribution	SD
one value	$N (μ, σ^{2})$	$σ$
the mean of $n$	$N (μ, σ^{2} / n)$	$σ / n$
the total of $n$	$N (n μ, n σ^{2})$	$n σ$

The first row needs a normal population. The second and third need a normal population or $n > 30$ (the CLT applies to a total exactly as to a mean, since $T = n \overset{ˉ}{X}$ ). Read the question for the words one, average/mean, or total/sum, write the row, and only then compute.

The paper's version, almost every sitting: parts (a) and (b) ask the same threshold about one value and about a mean, and part (c) asks why the answers differ. The answer is one sentence — the mean of $n$ varies far less than a single value, by the factor $n$ — and it is worth a mark on its own.

The three-way rule on N(120, 15²) with n = 25. The word in the question — one, average, total — picks the row; the row picks the SD.